Questions 1 What are main effects in ANOVA What are interactions in ANOVA How do you know you have an interaction What does it mean for a design to be completely crossed Balanced Orthogonal ID: 760063
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Slide1
Factorial ANOVA
2 or More IVs
Slide2Questions (1)
What are main effects in ANOVA? What are interactions in ANOVA? How do you know you have an interaction? What does it mean for a design to be completely crossed? Balanced? Orthogonal? Describe each term in a linear model like this one:
Slide3Questions (2)
Correctly interpret ANOVA summary tables. Identify mistakes in such tables. What’s the matter with this one?
Source
SS
Df
MS
F
A
512
2 (J-1)
128
128
B
108
1 (K-1)
108
54
AxB
96
2 (J-1)(K-1)
48
24
Error
12
5 N-JK
2
Slide4Questions (3)
Find correct critical values of
F
from a table
or software for
a given design.
How does post hoc testing for factorial ANOVA differ from post hoc testing in one-way ANOVA?
Describe a concrete example of a two-factor experiment. Why is it interesting and/or important to consider both factors in one experiment?
Slide52-way ANOVA
So far, 1-Way ANOVA, but can have 2 or more IVs. IVs aka Factors.Example: Study aids for examIV 1: workbook or notIV 2: 1 cup of coffee or not
Workbook (Factor A)
Caffeine (Factor B)
No
Yes
Yes
Caffeine only
Both
No
Neither (Control)
Workbook only
Slide6Main Effects (R & C means)
N=30 per cellWorkbook (Factor A)Row MeansCaffeine(Factor B)NoYesYesCaff =80SD=5Both =85SD=582.5NoControl =75SD=5Book =80SD=577.5Col Means77.582.580
Slide7Main Effects and Interactions
Main effects seen by row and column means; Slopes and breaks.Interactions seen by lack of parallel lines.Interactions are a main reason to use multiple IVs
Slide8Single Main Effect for B
(Coffee only)
Slide9Single Main Effect for A
(Workbook only)
Slide10Two Main Effects; Both A & B
Both workbook and coffee
Slide11Interaction (1)
Interactions take many forms; all show lack of parallel lines.
Coffee has no effect without the workbook.
Slide12Interaction (2)
People with workbook do better without coffee; people without workbook do better with coffee.
Slide13Interaction (3)
Coffee always helps, but it helps more if you use workbook.
Slide14Labeling Factorial Designs
Levels – each IV is referred to by its number of levels, e.g., 2X2, 3X2, 4X3 designs. Two by two factorial ANOVA.
Cell – treatment combination.
Completely Crossed designs –each level of each factor appears at all levels of other factors (vs. nested designs or confounded designs).
Balanced – each cell has same n.
Orthogonal design – random sampling and assignment to balanced cells in completely crossed design.
Slide15Review
What are main effects in ANOVA?
What are interactions in ANOVA? How do you know you have an interaction?
What does it mean for a design to be completely crossed? Balanced? Orthogonal?
Slide16Population Effects for 2-way
Population main effect associated with the treatment Aj (first factor):
Population main effect associated with treatment B
k
(second factor):
The interaction is defined as , so the linear model is:
The interaction is a residual:
Each person’s score is a deviation from a cell mean (error). The cell means vary for 3 reasons.
Slide17Expected Mean Squares
E(MS error) =
(Factor A has J levels; factor B has K levels; there are n people per cell.)
E
(MS A) =
E
(MS B) =
E
(MS Interaction) =
Note how all terms estimate error.
Slide18F Tests
For orthogonal designs, F tests for the main effects and the interaction are simple. For each, find the F ratio by dividing the MS for the effect of interest by MS error.
Effect
FdfAJ-1,N-JKBK-1,N-JKAxBInteraction(J-1)(K-1), N-JK
Slide19Example Factorial Design (1)
Effects of fatigue and alcohol consumption on driving performance.
Fatigue
Rested (8 hrs sleep then awake 4 hrs)
Fatigued (24 hrs no sleep)
Alcohol consumption
None (control)
2 beers
Blood alcohol .08 %
DV - performance errors on closed driving course rated by driving instructor.
Slide20Cells of the Design
Alcohol (Factor A)Orthogonal design; n=2Fatigue (Factor B)None(J=1)2 beers(J=2).08 %(J=3)Tired(K=1)Cell 12, 4 M=3Cell 216, 18M=17Cell 318, 20M=19Rested(K=2)Cell 40, 2M=1Cell 52, 4M=3Cell 616, 18M=17
M=2
M=10
M=18
M=13
M=7
M=10
Slide21Factorial Example Results
Main Effects?
Interactions?
Both main effects and the interaction appear significant. Let’s look.
Slide22Data
Person
DV
Cell
A alc
B rest
1
2
1
1
1
2
4
1
1
1
3
16
2
2
1
4
18
2
2
1
5
18
3
3
1
6
20
3
3
1
7
0
4
1
2
8
2
4
1
2
9
2
5
2
2
10
4
5
2
2
11
16
6
3
2
12
18
6
3
2
Mean
10
Slide23Total Sum of Squares
Person
DV
Mean
D
D*D
1
2
10
-8
64
2
4
10
-6
36
3
16
10
6
36
4
18
10
8
64
5
18
10
8
64
6
20
10
10
100
7
0
10
-10
100
8
2
10
-8
64
9
2
10
-8
64
10
4
10
-6
36
11
16
10
6
36
12
18
10
8
64
Total
728
Slide24SS Within Cells
Person
DV
Cell
Mean
D*D
1
2
1
3
1
2
4
1
3
1
3
16
2
17
1
4
18
2
17
1
5
18
3
19
1
6
20
3
19
1
7
0
4
1
1
8
2
4
1
1
9
2
5
3
1
10
4
5
3
1
11
16
6
17
1
12
18
6
17
1
Total
12
Slide25SS A – Effects of Alcohol
Person
M
Level (A)
Mean (A)
D*D
1
10
1
2
64
2
10
1
2
64
3
10
2
10
0
4
10
2
10
0
5
10
3
18
64
6
10
3
18
64
7
10
1
2
64
8
10
1
2
64
9
10
2
10
0
10
10
2
10
0
11
10
3
18
64
12
10
3
18
64
Total
512
Slide26SS B – Effects of Fatigue
Person
M
Level (B)
Mean (B)
D*D
1
10
1
13
9
2
10
1
13
9
3
10
1
13
9
4
10
1
13
9
5
10
1
13
9
6
10
1
13
9
7
10
2
7
9
8
10
2
7
9
9
10
2
7
9
10
10
2
7
9
11
10
2
7
9
12
10
2
7
9
Total
108
Slide27SS Cells – Total Between
Person
M
Cell
Mean (Cell)
D*D
1
10
1
3
49
2
10
1
3
49
3
10
2
17
49
4
10
2
17
49
5
10
3
19
81
6
10
3
19
81
7
10
4
1
81
8
10
4
1
81
9
10
5
3
49
10
10
5
3
49
11
10
6
17
49
12
10
6
17
49
Total
716
Slide28Summary Table
Source SSTotal728Between716A512B108Within12
Check: Total=Within+Between728 = 716+12
Interaction = Between – (A+B).Interaction = 716-(512+108) = 96.
SourceSSdfMSFA5122 (J-1)256128B1081(K-1)10854AxB962 (J-1)(K-1)4824Error126 (N-JK)2
Slide29Interactions
Choose the factor with more levels for X, the horizontal axis. Plot the means. Join the means by lines representing the other factor. The size of the interaction SS is proportional to the lack of parallel lines.If interactions exist, the main effects must be qualified for the interactions. Here, effect of alcohol depends on the amount of rest of the participant.
Slide30Software skills
Run the drink & drive problem in R
Data and code are in Canvas
Slide31Review
Correctly interpret ANOVA summary tables. Identify mistakes in such tables. What’s the matter with this one?
Source
SS
Df
MS
F
A
512
2 (J-1)
128
128
B
108
1 (K-1)
108
54
AxB
96
2 (J-1)(K-1)
48
24
Error
12
5 N-JK
2
Slide32Proportions of Variance
We can compute R-squared for magnitude of effect, but it’s biased, so the convention is to use omega-squared.
Slide33Planned Comparisons
CellMeanC1C21 A1B13-1/31/22 A2B117-1/3-1/43 A3B119-1/3-1/44 A1B211/31/25 A2B231/3-1/46 A3B2171/3-1/4-6-12B1 v B2 No alc v others
Slide34Planned Comparisons (2)
The first comparison (B1 v B2) has a value of –6. For any comparison,
Note this is the same as SS for B in the ANOVA.
Note 7.348 squared is 54, which is our value of F from the ANOVA.
(critical
t
has
df
e
)
Slide35Planned Comparisons (3)
We can substitute planned comparisons for tests of main effects; they are equivalent (if you choose the relevant means). We can also do the same for interactions. In general, there are a total of (Cells-1) independent comparisons we can make (6-1 or 5 in our example). Our second test compared no alcohol to all other conditions.
This looks to be the largest comparison with these data.
Slide36Post Hoc Tests
For post hoc tests about levels of a factor, we pool cells. The only real difference for Tukey HSD and Newman –Keuls is accounting for this difference. For interactions, we are back to comparing cells. Don’t test unless F for the effect is significant.
For comparing cells in the presence of an interaction:
Slide37Post Hoc (2)
In our example A has 3 levels, B has 2. Both were significant. No post hoc for B (2 levels). For A, the column means were 2, 10, and 18. Are they different?
A: Yes, all are different because the differences are larger than 3.07. But because of the interaction, the interpretation of differences in A or B are tricky.
Slide38Post Hoc (3)
For the rested folks, is the difference between no alcohol and 2 beers significant for driving errors? The means are 1 and 3.
A: They are not significantly different because 2 is less than 5.63. Note: data are fictitious. Do not drink and drive.
Slide39Review
Describe each term in a linear model like this one:How does post hoc testing for factorial ANOVA differ from post hoc testing in one-way ANOVA? Describe a concrete example of a two-factor experiment. Why is it interesting and/or important to consider both factors in one experiment?
Slide40Higher Order Factorials
If you can do ANOVA with 2 factors, you can do it with as many as you like.
For 3 factors, you have one 3-way interaction and three 2-way interactions.
Computations are simple but tedious.
For orthogonal, between-subject designs, all F tests have same denominator.
We generally don’t do designs with more than 3 factors. Complex & expensive.
Slide41Review R code
Download and run R code for driving (post hoc tests), run
chocolate example
Slide42Hays (5th ed) p. 520
A study involved children of three age groups:
A1=5yrs, A2= 6 years, A3 = 7 years.
Second Factor was formal preschool experience
B1=none, B2=1 year, B3 = more than 1 year
Five children were sampled from each combination and given a score on ‘social maturity’ (high scores are more mature)
Slide43B1
B2
B3
A1
2
2
4
A1
6
7
8
A1
8
8
11
A1
10
8
5
A1
9
10
3
A2
7
6
9
A2
9
9
14
A2
11
9
15
A2
9
7
10
A2
8
8
12
A3
15
13
15
A3
12
12
18
A3
9
10
21
A3
10
6
10
A3
14
9
14
Slide44Enter the data into R
Print the data to be sure they are correct
Be sure the data are arranged in the way
R
needs to analyze the problem
Slide45Run a 2-Way ANOVA on the data
What terms are significant
What is the interpretation at the level of main effects and interactions from the printout?
Slide46Graph the means
What is the interpretation of the means according to the graph?
Slide47What is the magnitude of effect
What is R-square for A, B, and A*B?
What is estimated omega-squared for each?
Slide48Unbalanced Designs
When you can assign to treatment, assign equal numbers in each cell
If unequal, unbalanced, and problems with tests
Participant mortality, design flaw, population characteristics (e.g., some types of cows more common in the herd)
Can delete or impute (but still problem)
Slide49All SS Types same if balanced
Type I SS – simply sequential in terms of model entry
(two-factor
study: A, B,
AxB
)
Test A; test B|A ; test
AxB|A
, B
Type II SS- partly sequential, considers terms at the same ‘level’ e.g.,
A, B test of A considers B but not
AxB
(A|B; B|A;
AxB|A
, B)
Type III SS – last in – regression SS
(A|B,
AxB
; B|A,
AxB
;
AxB|A
, B)
Dispute about best approach. SAS uses Type III, R uses Type II. Arguments are difficult to understand (hypotheses tested and model selection).